## Elliptical discrete solitons supported by enhanced photorefractive anisotropy

Optics Express, Vol. 16, Issue 6, pp. 3865-3870 (2008)

http://dx.doi.org/10.1364/OE.16.003865

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### Abstract

We demonstrate elliptical discrete solitons in an optically induced two-dimensional photonic lattice. The ellipticity of the discrete soliton results from enhanced photorefractive anisotropy and nonlocality under a nonconventional bias condition. We show that the ellipticity and orientation of the discrete solitons can be altered by changing the direction of the lattice beam and/or the bias field relative to the crystalline *c*-axis. Our experimental results are in good agreement with the theoretical prediction.

© 2008 Optical Society of America

## 1. Introduction

1. For a review, see
D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

2. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

7. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. **51**, 1520–1531 (1995). [CrossRef] [PubMed]

8. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. **36**, 419–424 (1996). [CrossRef]

11. O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett. **29**, 1248–1250 (2004). [CrossRef] [PubMed]

12. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. **29**, 268–270 (2004). [CrossRef] [PubMed]

14. P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop “Discrete solitons and vortices on anisotropic lattices,” Phys. Rev. E72, 046613 (2005). [CrossRef]

## 2. The theoretical model

*x*axis along one of the principal axes of the lattice beam as shown in Fig. 1. The angles of the external bias field and the crystalline

*c*-axis with respect to the

*x*axis are denoted by

*θ*and

_{e}*θ*, respectively. The probe and lattice beam propagate collinearly along the

_{c}*z*axis. Here, the polarization directions of the probe and lattice beam are always kept to be parallel and perpendicular to the

*c*-axis, respectively. Then the dimensionless equations governing the steady state propagation of the probe beam in the induced lattice can be written as [2

2. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

4. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, “Spatial solitons in optically induced gratings” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

7. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. **51**, 1520–1531 (1995). [CrossRef] [PubMed]

9. P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express **15**, 536–544 (2007). [CrossRef] [PubMed]

17. P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt. **45**, 2273–2278 (2006). [CrossRef] [PubMed]

*B*(

*x*,

*y*,

*z*)=

*b*(

*x*,

*y*)exp(i

*βz*), where

*β*is the propagation constant, and the real envelope

*b*(

*x*,

*y*) satisfies the following equation

## 3. Variation of induced lattice structures

*θ̣*corresponding to the three rows (form top to bottom) are 0, π/8, and π/4 (see the green arrows), respectively, and

_{c}*θ*for (a)–(e) are equal to

_{e}*θ*,

_{c}*θ*+π/4,

_{c}*θ*+π/2,

_{c}*θ*+3π/4, and

_{c}*θ*+π (see the purple arrows), respectively. The number given in the upper left corner of each figure corresponds to the index modulation depth of the lattice structure at

_{c}*E*

_{0}= 1. From Fig. 2, one can see that an identical lattice beam can induce different photonic lattice structures under various bias conditions (as shown in each row), and likewise for an identical bias condition, the lattice structures created by lattice beams with different orientations are distinct (as shown in each column). It should be noticed that under the same bias condition, the index modulation depth of the induced photonic lattice depends strongly on the orientations of the lattice beam. The bottom panels of Figs. 2(a) and 2(e) correspond to self-focusing lattices and self-defocusing backbone lattices under conventional bias conditions, as used in previous experiments of 2D discrete and gap solitons [3

3. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

5. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004). [CrossRef] [PubMed]

6. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides. “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90, 023902 (2003). [CrossRef] [PubMed]

15. B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B **86**, 399–405 (2007). [CrossRef]

*c*-axis. The induced photonic lattice structures under such a bias condition could lead to novel discrete self-localized states, as we shall demonstrate below.

## 4. Numerical results of discrete diffraction and self-trapping

*E*

_{0}= 1,

*Λ*= 8, and

*z*= 110, the output intensity pattern of the probe beam (for on-site excitation) after linear propagation is depicted in Fig. 3(a), where the panels from left to right correspond to the cases of

*θ*=

_{e}*θ*= π/4;

_{c}*θ*= 3π/4,

_{e}*θ*= π/4;

_{c}*θ*= 5π/8,

_{e}*θ*= π/8; and

_{c}*θ*= π/2,

_{e}*θ*= 0, respectively. Notice that, for comparison, we include one conventional bias case, i.e.,

_{c}*θ*=

_{e}*θ*= π/4. In the presence of nonlinearity, the input Gaussian beams can evolve into self-trapped states as shown in Figs. 3(b). It is clear that different conditions will lead to different patterns of linear diffractions and nonlinear self-trapped states. The soliton solutions with various peak intensities obtained by Petviashvili iteration method [18] for each case are displayed in Figs. 3(c) and 3(d), which shows that the solitons with higher peak intensities exhibit stronger localization. By comparing Figs. 3(b) and 3(c), we can see that the soliton profiles at lower peak intensities obtained by the nonlinear beam propagation and iteration method are very similar. Note that in the presence of photorefractive anisotropy, even under conventional bias condition, the soliton profile is somewhat elliptical (see the left column in Fig. 3). With enhanced anisotropy and nonlocality under

_{c}*E*

_{0}⊥

*c*, the ellipticity of the solitons will be dramatically increased, and the orientations of elliptical solitons can be altered by changing the relative orientations of lattice beam, bias direction, and the

*c*-axis. For the case with

*θ*= π/2 and

_{e}*θ*= 0, the equivalent lattice spacing is reduced while the index modulation is relatively low [see Fig. 2(c), top], thus the coupling between waveguide channels is much stronger, and both linear diffraction and nonlinear self-trapped state resembles that in the continuum limit (not well discretized).

_{c}## 5. Experimental results of discrete diffraction and self-trapping

5. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004). [CrossRef] [PubMed]

*c*-axis, respectively. A photorefractive SBN:60 crystal (with dimensions of 6.5mm×6.5mm×6.7(

*c*)mm and 0.025% by weight chromium dopant) is used. The intensity ratio between the probe and lattice beam is adjusted to be about 3:1. In our experiment, the lattice spacing is about 23µm. We first block off the probe beam until the lattice structure arrive steady-state, after that the linear and nonlinear transport of the probe beam is monitored simply by taking its instantaneous (before nonlinear self-action) and steady-state (after self-action) output patterns after propagating through the lattice. Because the induced lattice potential in our experiment is relatively shallow due to the weak absorption of the crystal sample at 532nm, thus a simulation using the experimental parameters is also performed for comparison.

*θ*=

_{e}*θ*= π/4;

_{c}*θ*= 3π/4,

_{e}*θ*= π/4; and

_{c}*θ*= 5π/8,

_{e}*θ*= π/8, respectively. The first and second rows in Figs. 4(a)–4(c) are for linear and nonlinear output beam patterns, respectively. The left column is the three-dimensional display of the middle column. The right column shows the simulation results obtained using our experimental parameters. The bias voltage for observations of Fig. 4(a) is 2kV and that for Figs. 4(b) and 4(c) is 3kV. It is obvious that under linear condition, the input beam undergoes discrete diffraction. While in the presence of nonlinearity, the probe beam can evolve into an elliptical self-trapped state. The observed patterns of discrete diffraction and elliptical solitons are in good agreement with those theoretical results obtained from the anisotropic photorefractive model. We also emphasize that, compared with the moving elliptical discrete solitons previous studied with an isotropic model [12

_{c}12. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. **29**, 268–270 (2004). [CrossRef] [PubMed]

## 6. Summary

*c*-axis, bias field, and lattice beam, we expect this can lead to a novel approach for band-gap engineering as well as diffraction and refraction management.

## Acknowledgments

## References and links

1. | For a review, see
D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E |

3. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

4. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, “Spatial solitons in optically induced gratings” Opt. Lett. |

5. | H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. |

6. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides. “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90, 023902 (2003). [CrossRef] [PubMed] |

7. | A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A. |

8. | A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. |

9. | P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, “Elliptical solitons in nonconventionally biased photorefractive crystals,” Opt. Express |

10. | E. D. Eugenieva, D. N. Christodoulides, and M. Segev, “Elliptic incoherent solitons in saturable nonlinear media,” Opt. Lett. |

11. | O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett. |

12. | J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. |

13. | F. Ye, L. Dong, J. Wang, T. Cai, and Y. Li, “Discrete elliptic solitons in two-dimensional waveguide arrays,” Chin. Opt. Lett. |

14. | P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop “Discrete solitons and vortices on anisotropic lattices,” Phys. Rev. E72, 046613 (2005). [CrossRef] |

15. | B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B |

16. | P. Zhang, S. Liu, J. Zhao, C. Lou, J. Xu, and Z. Chen, “Optically induced transition between discrete and gap solitons in a nonconventionally biased photorefractive crystal,” to appear in Opt. Lett. (2008). [CrossRef] [PubMed] |

17. | P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt. |

18. | V. I. Petviashvili, “Equation of an extraordinary soliton,” Sov. J. Plasma Phys. |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 25, 2008

Revised Manuscript: February 27, 2008

Manuscript Accepted: February 27, 2008

Published: March 10, 2008

**Citation**

Peng Zhang, Jianlin Zhao, Fajun Xiao, Cibo Lou, Jingjun Xu, and Zhigang Chen, "Elliptical discrete solitons supported by enhanced photorefractive anisotropy," Opt. Express **16**, 3865-3870 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3865

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### References

- For a review, see D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002). [CrossRef]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003). [CrossRef] [PubMed]
- H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides. “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
- A. A. Zozulya and D. Z. Anderson, "Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field," Phys. Rev. A 51, 1520-1531 (1995). [CrossRef] [PubMed]
- A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Self-focusing and soliton formation in media with anisotropic nonlocal material response," Europhys. Lett. 36, 419-424 (1996). [CrossRef]
- P. Zhang, J. Zhao, C. Lou, X. Tan, Y. Gao, Q. Liu, D. Yang, J. Xu, and Z. Chen, "Elliptical solitons in nonconventionally biased photorefractive crystals," Opt. Express 15, 536-544 (2007). [CrossRef] [PubMed]
- E. D. Eugenieva, D. N. Christodoulides, and M. Segev, "Elliptic incoherent solitons in saturable nonlinear media," Opt. Lett. 25, 972-974 (2000). [CrossRef]
- O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, "Observation of elliptic incoherent spatial solitons," Opt. Lett. 29, 1248-1250 (2004). [CrossRef] [PubMed]
- J. Hudock, N. K. Efremidis, and D. N. Christodoulides, "Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays," Opt. Lett. 29, 268-270 (2004). [CrossRef] [PubMed]
- F. Ye, L. Dong, J. Wang, T. Cai, and Y. Li, "Discrete elliptic solitons in two-dimensional waveguide arrays," Chin. Opt. Lett. 3, 227-229 (2005).
- P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop "Discrete solitons and vortices on anisotropic lattices," Phys. Rev. E 72, 046613 (2005). [CrossRef]
- B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, "Anisotropic photonic lattices and discrete solitons in photorefractive media," Appl. Phys. B 86, 399-405 (2007). [CrossRef]
- P. Zhang, S. Liu, J. Zhao, C. Lou, J. Xu, and Z. Chen, "Optically induced transition between discrete and gap solitons in a nonconventionally biased photorefractive crystal," to appear in Opt. Lett. (2008). [CrossRef] [PubMed]
- P. Zhang, Y. Ma, J. Zhao, D. Yang, and H. Xu, "One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal," Appl. Opt. 45, 2273-2278 (2006). [CrossRef] [PubMed]
- V. I. Petviashvili, "Equation of an extraordinary soliton," Sov. J. Plasma Phys. 2, 257-258 (1976).

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